**The Laplace Transform**

Part 18: The Bessel Functions

We can use the differential equation technique demonstrated in the previous post to find the Laplace transforms of the Bessel function of the first kind *J*_{n}(*t*) for nonnegative integer order *n*. These are defined as the solutions to the Bessel differential equation:

which are non-singular at the origin. Specifically, for integer *n*≥0, .

First, let’s find the transform of *J*_{0}(*t*). This is a solution to with *n*=0, so

,

or dividing by *t*,

.

Taking the Laplace transform, and using we have

,

and we have a first order linear differential equation for the transform *Y*(*s*). In fact, this equation is separable:

,

where *C* is a constant, arising from our integration. To find its value, consider the initial value theorem:

.

We see

,

and so . Thus, we see that the transform of the Bessel function of the first kind of order zero is

.

Now, to find *n*=1, we use a derivative identity for the Bessel functions of the first kind (equation number 59 here):

.

Using the product rule on the left-hand side, we see

.

Taking the Laplace transform, and letting , we have

.

Letting *n*=1, we then get

.

Integrating,

.

To find the constant of integration, we consider the limit as *s*→∞; we saw earlier that , so to get , required for *J*_{1}(*t*) to be nonsingular for *t*=0, we must have *C*=1, and so

.

In general, one can use the relation and proof by induction to show that the general formula is

,

for integer *n*≥0.

Using the scaling rule and some algebra, this generalizes to

.

### Like this:

Like Loading...

*Related*

Tags: Bessel Functions, Differential Equation, Laplace Transform, Math, Monday Math

This entry was posted on May 3, 2010 at 4:24 am and is filed under Math/Science, Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

## Leave a Reply